Riemann theta functions

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1: 21.2 Definitions

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§21.2(i) Riemann Theta Functions

… ►For numerical purposes we use the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … ►

§21.2(ii) Riemann Theta Functions with Characteristics

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§21.2(iii) Relation to Classical Theta Functions

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2: 21.8 Abelian Functions

§21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

3: 21.10 Methods of Computation

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§21.10(i) General Riemann Theta Functions

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§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface

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Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

4: 21.9 Integrable Equations

§21.9 Integrable Equations

… ►Typical examples of such equations are the Korteweg–de Vries equation … ► … ►

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Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify

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5: 21.3 Symmetry and Quasi-Periodicity

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§21.3(i) Riemann Theta Functions

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§21.3(ii) Riemann Theta Functions with Characteristics

… ► …For Riemann theta functions with half-period characteristics, …

6: 21.1 Special Notation

… ►Uppercase boldface letters are $g\times g$ real or complex matrices. ►The main functions treated in this chapter are the Riemann theta functions $\theta \left(𝐳\right|𝛀)$, and the Riemann theta functions with characteristics $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$. ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).

7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►Such a solution is given in terms of a Riemann theta function with two phases. …

8: 21.4 Graphics

§21.4 Graphics

►Figure 21.4.1 provides surfaces of the scaled Riemann theta function $\widehat{\theta }\left(𝐳\right|𝛀)$, with … ►For the scaled Riemann theta functions depicted in Figures 21.4.2–21.4.5 … ►

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Figure 21.4.4: A real-valued scaled Riemann theta function: $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|{𝛀}_1)$, $0\le x\le 4$, $0\le y\le 4$. … Magnify 3D Help

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. … Magnify 3D Help

9: 21.5 Modular Transformations

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§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. ►The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which $\xi (𝚪)$ is determinate: … ►

§21.5(ii) Riemann Theta Functions with Characteristics

… ►For explicit results in the case $g=1$, see §20.7(viii).

10: 21.6 Products

§21.6 Products

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21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

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http://dlmf.nist.gov/21.6.E3

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See also:

Annotations for §21.6(i), §21.6 and Ch.21

… ►On using theta functions with characteristics, it becomes … ►

§21.6(ii) Addition Formulas

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